MAADHYAM Nurturing Gifted Minds
INSIDE HIGHLIGHTS:
Mathematics Behind Pyramid and Taj Mahal
Exploring mathematically designed buildings around the
world
Why architects use triangular structures?
Maadhyam News Corner
MONTH:NOVEMBER ISSUE NO:2015(4)
Printed under Gifted Education Abhiyaan
An initiative by the office of principal Scientific
Advisor to the Government of India
Have you ever wondered how buildings are constructed?
Are our buildings shaped by
sacred numbers and hidden
codes?
What factors determine the
maximum height of a
monument? ?
Is there any building which is
mathematically designed?
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"In mathematics, if a pattern occurs we can go on and ask 'Why does it occur?' 'What does it signify?' and we can find answers to these questions." W.W.Sawyer Mathematics is full of beautiful patterns. Sacred geometry, or spiritual geometry, is the belief that numbers and patterns such as the divine ratio have sacred significance. Many mystical and spiritual practices, including astrology, numerology, tarot, and feng shui, begin with a fundamental belief in sacred geometry. Architects and designers may draw upon concepts of sacred geometry when they choose particular geometric forms to create pleasing, soul-satisfying spaces. Architecture begins with geometry. Since earliest times, architects have relied on mathematical principles. The ancient Roman architect Marcus Vitruvius Pollio believed that builders should always use precise ratios when constructing temples. "For without symmetry and proportion no temple can have a regular plan,“ From the pyramids in Egypt to the new World Trade Center tower in New York, good architects use the same essential building blocks as your body and all living things. Moreover, the principles of geometry are not confined to great temples and monuments. Mathematicians say that when we recognize geometric principles and build upon them, we create dwellings that comfort and inspire.
Mathematics In Architecture
To read more about M.V.Pollio, visit
http://www.vitruvius-pollio.com/
Strange Before you dismiss the idea of sacred geometry, take a few moments to reflect on the ways some numbers and patterns appear time and again in every part of your life.
MONTH:NOVEMBER ISSUE NO:2015(4)
The Pythagorean theorem and the 3-4-5 right triangle was used by rope stretchers and Egyptian
engineers. Rope was knotted into 12 sections that stretched out to produce a 3-4-5 triangle. For e.g. If
you tie 12 equally spaced knots into a rope, fix the rope at the 5th knot to the ground, go with the longer
side 4 knots in one direction, fix the rope there too, and then try to bring both lose ends together, you
will automatically get a right angle. The Egyptians also used length/height-ratios to construct pyramid
slopes. Mostly used were 1:22 and 1:21-ratios, one cubit height to 21 or 22 fingers length. If you
substitute the cubit with the fingers you get for Kephrens pyramid a 28:21-ratio. Divide this by 7, and
you get a "3 units length by 4 units height"-ratio. These are the short sides of a holy triangle, so the third
side must then be in a "5 units"-ratio in respect to the other sides. Therefore, all pyramids in Egypt
constructed in the 1:21-ratio are carrying the 3:4:5-ratio quite naturally.
The Great Pyramid was laid out with geometric precision - a near-perfect square base, with sides of 230
meters that differ from each other by less than twenty centimeters, and faces that sloped upwards at an
angle of 51º to reach an apex nearly 150 meters above the desert floor.
Some people have made certain discoveries about the Great Pyramid, using maths:
When using the Egyptian cubit the perimeter is 365.24 - the amount of days in the year.
The height x 10 to the power of 9 gives approximately the distance from the earth to the sun.
The perimeter divided by 2 x the height of the pyramid is equal to pi - 3.1416
The weight of the pyramid x 10 to the power of 15 is equal to the approximate weight of the earth.
When the cross diagonals of the base are added together, the answer is equal to the amount of time
(in years) that it takes for the earth's polar axis to go back to its original starting point - 25,286.6
years.
The link between mathematics and architecture goes back to ancient times, when the two disciplines were virtually
indistinguishable. Pyramids and temples were some of the earliest examples of mathematical principles at work. Today,
mathematics continues to feature prominently in building design.. We are not just talking about mere measurements, — though elements like that are integral to architecture.
Thanks to modern technology, architects can explore a variety of exciting design options based on complex mathematical languages, allowing them to build groundbreaking forms.
Now take a look at several structures in the past that were modeled along mathematics.
DO YOU KNOW ???
The Great Pyramid of Giza in 4700
B.C. is with proportions according to
a “sacred ratio”. The ancient
Egyptians constructed the Great
Pyramids in such a way that the ratio
(b : h : a) is approximately equal to
(1 : √φ : φ).
To know more on mathematical proof visit the link: https://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html;http://www.halexandria.org/dward106.html 2
MONTH:NOVEMBER ISSUE NO:2015(4)
Mathematics in Taj Mahal Symmetry is found in the world around us: in nature, in artwork and even in buildings we see everyday.
You can see it in structures such as Taj Mahal in India. You might not even realize it, but you probably
like images that are symmetrical. Designers, architects, and artists understand this, which is why they
often use symmetry to create images that are pleasing to us.
The next time you visit the Taj Mahal in India, waiting to get that iconic photo in front of this beautiful
building. Look closer and you’ll find a great example of line symmetry – with two lines, one vertical
down the middle of the Taj, and one along the waterline, showing the reflection of the prayer towers in
the water. The Taj Mahal (completed in 1648) exhibits symmetry, a mathematical property claims that
Divine Proportion was used in the construction of the Taj Mahal.
The Taj Mahal displays golden proportions in the width of its grand central arch to its width, and also
in the height of the windows inside the arch to the height of the main section below the domes.
Divine Proportion...
What is this?
Golden ratio, also known as the golden section, golden mean,
or divine proportion, in mathematics, the irrational number
(1 + √5)/2, often denoted by the Greek letters τ or ϕ, and
approximately equal to 1.618.
The origin of this number and its name may be traced back to
about 500 BC and the investigation in Pythagorean geometry
of the regular pentagon.
We find the golden ratio when we divide a line into two parts
so that: the longer part divided by the smaller part is also
equal to the whole length divided by the longer part.
In terms algebra, letting the length of the shorter segment be
one unit and the length of the larger segment be x units gives
rise to the equation (x + 1)/x = x/1; this may be rearranged to
form the quadratic equation X2 – x – 1 = 0, for which the
positive solution is x = (1 + √5)/2, the golden ratio.
To know more about Golden Ratio visit
the given links: 1. https://www.youtube.com/watch?v=lm9zWqJ6-Rg;
2.https://www.youtube.com/watch?v=1_dDMdPjt60,
3. https://www.mathsisfun.com/numbers/golden-
ratio.html;
4. http://www.basic-mathematics.com/what-is-the-
golden-ratio.html;
5. http://mathart.wikidot.com/golden-ratio2
6. http://www.cut-the-
knot.org/do_you_know/GoldenRatio.shtml
We will now prove that the ratio of the lengths of
two diagonals is indeed the Golden ratio.
Assume that rectangle ABCD is a Golden
Rectangle.
Hence, AD/AB =AE/ED. But, FE = AE, and so
FE/ED=.
Hence, rectangle FCDE is a Golden Rectangle.
We have two similar rectangles and so since =
AD/EF then BD/CE = .
An interesting thing happens when we work with
these rectangle. Suppose we take a rectangle of side 1
unit and a rectangle of side 2 units and we put them
side to side in the following way and draw the
appropriate segments to form a rectangle. If we
continue to create rectangles in this way we will get a
series of rectangles like in figure. The following
picture shows several such rectangles, and the lengths
of their sides.
If we take ratios of the length we will see that the series of
whirling rectangles will begin to estimate the Golden Ratio.
2/1 = 2 3/2= 1.5 5/3 = 1.666... 8/5 = 1.6 13/8 = 1.625 and so on.
Hence as we increase the number of squares we get a figure that
begins to look more and more like the Golden Rectangle. It might
also be noticed that there is something special about the sides of
the squares. If we list them we have, 1, 2, 3, 5, 8, 13, ... This of
course is the famous Fibonacci sequence. Visit the link for
mathematical proof http://www.basic-mathematics.com/what-is-
the-golden-ratio.html
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Image Source: www.phimatrix.com
MONTH:NOVEMBER ISSUE NO:2015(4)
The British physicist and mathematician, Roger Penrose, developed a periodic tiling which incorporates the golden section. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B). When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the “Golden Ratio." In addition to the unusual symmetry, Penrose Tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean. Visit link :
http://www.ams.org/samplings/feature-column/fcarc-penrose
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Geometrical Construction of Golden Ratio
Now we will construct the Golden Rectangle.
1) First construct a square ABCD. 2) Construct the midpoint E of DC. 3) Extend DC. With center E and radius EB, draw an arc crossing EC extended at C. 4) Construct a perpendicular to DF at F. 5) Extend AB to intersect the perpendicular at G. 6) AGFD is a Golden Rectangle. 7) Now measure the length and width of the rectangle. Then find the ratio of the
length to the width. This should be close to the Golden Ratio (approximately 1.618).
You can also find golden ratio in
Modern abstract art such as Penrose
Tilings.
MONTH:NOVEMBER ISSUE NO:2015(4)
Parthenon, Athens, Greece
•Constructed in 430 or 440 BC the Parthenon was built on the Ancient Greek ideals of harmony, demonstrated by the building’s perfect proportions.
•The width to height ratio of 9:4 governs the vertical and horizontal proportions of the temple as well as other relationships of the building, for example the spacing between the columns.
•It’s also been suggested that the Parthenon’s proportions are based on the Golden Ratio (found in a rectangle whose sides are 1: 1.618).
Sydney Harbour Bridge
•The Sydney Harbour bridge is a magnificent structure of mathematical genius, located in what has to be the world’s most beautiful city.
•The mathematics associated with the Sydney Bridge, including deriving the Quadratic Equations for both the lower and upper parabolic arches of the bridge.
•Read more for mathematical proof of bridge: http://passyworldofmathematics.com/sydney-harbour-bridge-mathematics/
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Taj Mahal
Sydney Harbour Bridge
Akshardham
- a 3D fractal like
temple
The Great
Pyramid of Giza
Morbius Strip Temple
Magic Square
Cathedral Gherkin's curves, London
The Eden Project, Cornwall,
UK
Parthenon
Cube Village
Experimental Math-Music
Pavilion
MONTH:NOVEMBER ISSUE NO:2015(4)
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Bahá'í-House of worship, New Delhi
The beautiful concept of the lotus, as conceived by the architect, had to be converted into definable geometrical shapes such as spheres, cylinders, toroids and cones. These shapes were translated into equations, which were then used as a basis for structural analysis and engineering drawings.
Unlike conventional structures for which the elements are defined by dimensions and levels, here the shape, size, thickness, and other details were indicated in the drawings only by levels, radii, and equations.
The resultant geometry was so complex that it took the designers over two and a half years to complete the detailed drawings of the temple.
Follow the link to know more: http://www.bahaihouseofworship.in/architectural-blossoming
The Eden Project, Cornwall, UK
•It’s little surprise that the building has taken its inspiration from plants, using Fibonacci numbers(0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... where every number is the sum of the previous two) to reflect the nature featured within the site.
•The outer layer is made of hexagons (the largest is 11 metres across), plus the odd pentagon. The inner layer comprises hexagons and triangles bolted together.
•The geodesic concept provided for least weight and maximum surface area on the curve – with strength.
•Read more: https://www.edenproject.com/eden-story/behind-the-scenes/architecture-at-eden
Mobius Strip Temple
•The Mobius Strip is a surface with only one side and only one boundary.
• An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a loop.
•The Möbius strip has the mathematical property of being non-orientable, while the Möbius strip has several curious properties.
•A line drawn starting from the seam down the middle will meet back at the seam but at the “other side”. If continued the line will meet the starting point and will be double the length of the original strip.
•This single continuous curve demonstrates that the Möbius strip has only one boundary.
•Read more : http://www.arch2o.com/buddhism-temple-miliy-design/
MONTH:NOVEMBER ISSUE NO:2015(4)
Have you noticed that many
architectures have used triangular
structures!
Yes true! But why are
triangles used in construction of buildings?
As we have read, the most famous triangular structures are, of
course, The Great Pyramids of Egypt and South America. The
strongest part of a pyramid is the wide base. Each successive row has
less weight to support above it.
And also some of these monumental
tetrahedrons have been standing for tens of thousands of years.
A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed.
Properties:
Interior angles of triangles (angles on the inside) sum up to 180°.
Triangle Inequality Theorem
Relationship between measurement of the sides and angles in a Triangle
To know more in detail visit http://www.mathwarehouse.com/geometry/triangles/
Triangle does not easily deform and is able to balance the stretching
and compressive forces inside the structure.
For economic reasons: since the triangle obviously has only 3 sides, it
requires little material to make a support, thus minimizing the costs.
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Why are triangles so strong?
A square lacks the
rigid strength of a
triangle.
But by adding diagonal
bracing, a common feature in
bridges and buildings, the
structure can again rely on the
strength of a triangle to hold
its shape.
Try Yourself
Triangles are inherently strong because they form a fixed rigid
shape.
This can be demonstrated by building a triangle out of garden canes,
securing the corners with rubber bands. The shape is fixed by the
length of the sides and the triangle withstands quite substantial forces
applied to it.
However if you built a square in the same way, a gentle push at one
corner could easily change the shape into a parallelogram. There are
infinitely many four-sided shapes with equal sides, a square is just one
of them, and so the shape is easily transformed from one to the other
with minimal force.
What could you conclude from this?
To know more
Visit the links:
http://www.pbs.
org/wgbh/build
ingbig/bridge/b
asics.html
http://www.fac
ulty.fairfield.ed
u/jmac/rs/bridg
es.htm
MONTH:NOVEMBER ISSUE NO:2015(4)
M.C.ESCHER Art and Math may at first seem to be very different things, but people who enjoy math tend to look for mathematics in art. They want to see the patterns and angles and lines of perspective. This is why artists like M.C. Escher appeal to mathematicians so much. There is a large amount of math involved in art, not to mention basic things like measuring and lines, but the intricacies of art can often be described using math. Visit the given link to know more about M.C. Escher http://www.mcescher.com/about/ Escher is a famous artist who created mathematically challenging artwork. He used only simple drawing tools and the naked eye, but was able to create stunning mathematical pieces. He produced polytypes, sometimes in drawings, which cannot be constructed in the real world, but can be described using mathematics. His drawings caught the eyes and looked possible by perception, but were mathematically impossible. His particular drawing, Ascending and Descending, was one of the masterpieces. In this drawing, Escher creates a staircase that continues to ascend and descend, which is mathematically impossible, but the drawing makes it seem realistic.
Let’s Know About
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Relativity
Image Source:
http://www.mcescher.com/gallery/back-in-holland/relativity/
M.C. Escher ~ Cycle Art Print
Image Source: www.leninimports.com
Tessellating Prints - An Introduction - David Bailey s
World of Escher-like Tessellations.
Image Source: http://phoenix-588-3.iaafmt.org/
Mathematical Tessellation- Shells and Starfish
Image Source: http://urioste.weebly.com/tessellations.html
MONTH:NOVEMBER ISSUE NO:2015(4)
Noida boy makes India proud, wins biggest
maths puzzle championship in US Published:
Thursday, June 11, 2015, 17:50 [IST]
Noida's Gaurav Pandey doubled the celebration by
becoming KenKen international champion. KenKen
is a grid-based numerical puzzle that uses basic
math operations while challenging the logic and
problem-solving skills of participants. It is
recognised by NCTM (National Council of
Teachers of Mathematics, USA), an independent
authority, as a powerful tool to build reasoning
skills in kids.
Read more at:
http://www.oneindia.com/india/noida-boy-
makes-india-proud-wins-biggest-maths-puzzle-
championship-in-us-1774508.html
Indian-origin mathematician Manjul
Bhargava wins Fields Medal
Manjul Bhargava was awarded the prestigious
medal for ‘developing powerful new methods in
geometry of numbers’. Manjul Bhargava, a
Canadian mathematician of Indian origin, has
been awarded the prestigious 2014 Fields Medal
at the International Mathematical Union’s (IMU)
International Congress of Mathematicians held
in Seoul.
Read more at:
http://www.livemint.com/Politics/76RVvYH
Nx7neqcW1gEmCNN/Indianorigin-
mathematician-Manjul-Bhargava-awarded-
Fields-M.html
Visit the following link to know about latest
updates on mathematics and science
https://plus.maths.org/content/News
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Maadhyam News Corner
2014 Nobel Prize in Chemistry: Super-resolved fluorescence microscopy
Life on Earth likely started 4.1 billion years ago,
much earlier than scientists thought
University of California -Los Angeles geochemists
have found evidence that life likely existed on Earth
at least 4.1 billion years ago -- 300 million years
earlier than previous research suggested. The
discovery indicates that life may have begun shortly
after the planet formed 4.54 billion years ago.
To know more:
http://www.sciencedaily.com/releases/2015/10/1
51019154153.htm
http://universityofcalifornia.edu/
2014 Nobel Prize in Chemistry: Super-
resolved fluorescence microscopy
The 2014 Nobel Prize in Chemistry has been
awarded to Eric Betzig of Janelia Farm
Research Campus, Howard Hughes Medical
Institute; Stefan W. Hell of Max Planck Institute
for Biophysical Chemistry and the German
Cancer Research Center; and William E.
Moerner of Stanford University "for the
development of super-resolved fluorescence
microscopy."To know more:
http://www.sciencedaily.com/releases/2014/1
0/141008085419.htm
http://www.kva.se/en/ Deep-sea bacteria could help neutralize
greenhouse gas
A type of bacteria plucked from the bottom of the
ocean could be put to work neutralizing large
amounts of industrial carbon dioxide in the Earth's
atmosphere, a group of University of Florida
researchers has found.
To know more:
http://www.sciencedaily.com/releases/2015/10/1
51022141716.htm
http://www.ufl.edu/
MONTH:NOVEMBER ISSUE NO:2015(4)
Response Sheet
Q1: Consider the following diagram,
Given that the areas of A1, A2 and A3 in the diagram
are equal, show that 𝑅𝑋
𝑋𝑆=
𝑅𝑌
𝑌𝑄=
5 2+1
2
so that the points X and Y divide the sides of the rectangle
in the golden ratio.
____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ Q2: Why are quadrilaterals unstable? Compare their stability with triangular structures.
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Q3.In order to test how strong a triangular structure is, Atul has done an experiment.
First of all, He made an accordion fold of an A4 sheet of paper.
Used two books as supporters and put the paper on top of them.
Now, He put a load on it. What can you conclude from this?
Will the A4 paper be able to support two books which are far heavier
than its own weight or not? Give reason.
___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________
Q4.
The diagram shows a rhombus PQRS with an internal point O such that OQ=OR=OS=1unit. Penrose used this rhombus, split into two quadrilaterals, a dart and a kite, to make his famous tiling which fills the plane but, unlike a tessellation, does not repeat itself by translation or rotation.
Find all the angles in the diagram, show that POR is a straight line and show that triangles PRS and QRO are similar. Hence prove that the length of the side of the rhombus is equal to the Golden Ratio.
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MONTH:NOVEMBER ISSUE NO:2015(4)
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RESEARCH TEAM
Geetu Sehgal , Shilpi
Bariar, Jyoti Batra,
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Verma,Meenakshi
Gifted Education Abhiyaan
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